Learning Objectives
- Explain the concept of the Kalman filter.
- Describe the fundamentals of measurement and measurement uncertainty.
- Justify the necessity of prediction in sensing.
- Explain the fundamental terms such as variance, standard deviation, normal distribution, estimate, accuracy, precision, mean, expected value, and random variable.
Do you play sports like table tennis, badminton or pickled ball?
- We use our eyes to detect the ball, but our brain needs to project its trajectory and hit it seamlessly
another example:
Air Defense System - shooting down Fighter Jet
- Do you think that just detecting is enough in this case?
-
Suppose we have a track cycle of 5 seconds. At intervals of 5 seconds, the radar
samples the target by directing a dedicated pencil beam. -
The future target position can be easily calculated using Newton’s motion equations:
Where: -
xis the target position -
x₀is the initial target position -
v₀is the initial target velocity -
ais the target acceleration -
Δtis the time interval (5 seconds in our example) -
As for 3D:
-
The set of target parameters
The 3D motion equations can be written in state-space form:
\begin{bmatrix}
x{k+1} \
y{k+1} \
z{k+1} \
v{x, k+1} \
v{y, k+1} \
v{z, k+1}
\end{bmatrix}
=
\begin{bmatrix}
1 & 0 & 0 & \Delta t & 0 & 0 \
0 & 1 & 0 & 0 & \Delta t & 0 \
0 & 0 & 1 & 0 & 0 & \Delta t \
0 & 0 & 0 & 1 & 0 & 0 \
0 & 0 & 0 & 0 & 1 & 0 \
0 & 0 & 0 & 0 & 0 & 1 \
\end{bmatrix}
\begin{bmatrix}
xk \
y_k \
z_k \
v{x,k} \
v{y,k} \
v{z,k}
\end{bmatrix}
- \begin{bmatrix}
\frac{1}{2} \Delta t^2 & 0 & 0 \
0 & \frac{1}{2} \Delta t^2 & 0 \
0 & 0 & \frac{1}{2} \Delta t^2 \
\Delta t & 0 & 0 \
0 & \Delta t & 0 \
0 & 0 & \Delta t \
\end{bmatrix}
\begin{bmatrix}
a{x,k} \
a{y,k} \
a_{z,k}
\end{bmatrix}
$$
- In reality, the radar measurement is not entirely accurate. It contains random
errors or uncertainties that can affect the accuracy. This random errors or
uncertainties in the radar measurement are known as Measurement Noise. - In addition, the target motion is not always aligned with the motion equations due
to external factors like wind, air turbulence, and pilot maneuvers. This misalignment
between the motion equations and the actual target motion results in an error or
uncertainty in the dynamic model, which is called Process Noise. - Due to the Measurement Noise and the Process Noise, the estimated target position
can be far away from the actual target position. In this case, the radar might send
the track beam in the wrong direction and miss the target. - In order to improve the radar’s tracking accuracy, it is essential to employ a prediction
algorithm that accounts for both process and measurement uncertainty. - The most common tracking and prediction algorithm is the Kalman Filter.
Essential background
Mean
Example:
- Now assume five different weight measurements of the same person: 79.8kg, 80kg,
80.1kg, 79.8kg, and 80.2kg. The person is a system, and the person’s weight is a
system state.
Variance and Standard deviation
- The Variance is a measure of the spreading of the data set from its mean.
- The Standard Deviation is the square root of the variance.
- The standard deviation is denoted by the Greek letter
Example:
- Suppose we want to compare the heights of two high school basketball teams.
- Calculate the distance from the mean for each variable by subtracting the mean from each variable.
Some of the values are negative. To get rid of the negative values, let us square the
distance from the mean:
To calculate the variance of the data set, we need to find the average value of all
squared distances from the mean:
- We can see that although the mean of both teams is the same, the measure of the
height spreading of Team A is higher than the measure of the height spreading of Team B. Therefore, the Team A players are more diverse than the Team B players.
-The units of the variance are meters squared; it is more convenient to look at the
standard deviation, which is a square root of the variance.
• The standard deviation of Team B players’ heights would be 0.036m.
Normal Distribution
- The Gaussian curve is also called the PDF (Probability Density Function) for
the normal distribution.
Example:
- The following chart describes PDFs of the pizza delivery time in three cities: city
’A,’ city ’B,’ and city ’C.’
• In city ’A,’ the mean delivery time is 30 minutes, and the standard deviation
is 5 minutes.
• In city ’B,’ the mean delivery time is 40 minutes, and the standard deviation
is 5 minutes.
• In city ’C,’ the mean delivery time is 30 minutes, and the standard deviation
is 10 minutes.
- 68.26% of the pizza delivery times in City A lie within μ ± σ range (25-35 minutes)
- 95.44% of the pizza delivery times in City A lie within μ ± 2σ range (20-40 minutes)
- 99.74% of the pizza delivery times in City A lie within μ ± 3σ range (15-45 minutes)
Estimate, Accuracy and Precision
Additional notes:
Random Variables
1. Raw Moments (also called uncentered moments)
-
The kth raw moment is:
That is, the expected value (mean) of the kth power of the random variable XXX.Examples:
- First raw moment:
- Second raw moment:
- First raw moment:
2. Central Moments
-
The kth central moment is:
where
This measures how much the values of XXX deviate from the mean, raised to the power
Examples:
- Second central moment:
- Third and fourth central moments are used to study skewness and kurtosis (asymmetry and peak shape) of the distribution.
- Second central moment: